I've been thinking about game theory-ish situations earlier at work today. Hopefully some of you can validate my thinking or correct me where I'm wrong.





Game #1
Here is a game where two players are each dealt a number 0-100 where neither player sees the other person's card. There is $0.50 anted from both players making a pot total of $1. There is a round of betting where player 1 bets $1 or folds. If player 1 decides to bet then player 2 has the option to call or fold.

According to game theory player 2 needs to call with 50% of his hands to keep player 1 indifferent to bluffing. player 1 must bet 1 bluff for every 2 value hands to make player 2 indifferent to calling. Not sure if I got this correct or not, but I believe that a "value" hand in GT world is any hand that beats the bottom of player 2's calling range.

So player 1 bets his top 50% hands for value and bluffs 25% of other hands for bluffs. These hands don't really matter, but just for the sake of it we'll make it numbers from 25 to 50. All-in all, player 1 bets 75% of hands and player 2 calls the top 50% of hands


Ranges:
Player 1 bets 100-25; and folds 25-0
Player 2 calls 100-50; and folds 50-0


Questions:
1A) Is my definition of a value hand correct?
1B) What is player 1 and player 2's EV in this game?
1C) How does the equilibrium change if Player 1's options are checking instead of folding or is it only the EV that changes (assuming that checking closes the action)?





Game #2
Now to change things up player 2 has the option to raise to $4 when facing a bet. Player 1 will then have the option to call or fold. I'm not sure if this effects player 1's betting range a bit, but for now lets assume it doesn't.

Player 1 bets 75% of hands. Player 1 will call a re-raise with top 33% of the 75% since player 1 is risking $4 to win $2. Calling 33% keeps player 2's raises indifferent. Top 33% of the 75% is top 24.75% of hands.

This means Player 2 must raise with his value hands which is also 24.75% of hands, but also including 1/2 of that for bluffs. so 24.75%*(1/2) + 24.75% = 37.125% of hands total. Player 2 is going to want to bluff hands that he can't call with, because his calling range will perform better if stuff isn't taken from it.


Ranges:
Player 1 bets 100-25; and folds 25-0
Player 2 bets 100-75.25 and 50-37.625; calls 75.25-50; folds 37.265-0
Player 1 calls 100-75.25 and folds 75.25-25


Questions:
2A) Does the fact that Player 2 can raise effect Player 1's opening range %?
2B) If we continue the pattern where we keep allowing the last person to raise yet again, does the last person that gets the raise always have the advantage or is it an exponential decay of added value? (I hope you get what I mean, but I'm having a hard time wording it).





Game #3
Take the same rules from Game 1, but give player 1 the option to check. After checking the action is NOT closed. Player 2 then has the option to check (closing the action) or bet $1. Player 1 can then either call or fold.

I'm assuming that the betting strategies for player 1 is more or less the same. He will probably bluff with the bottom part of his entire range and check with the slightly higher crap. However, he will be forced to check hands that will be strong enough to not let player 2 bet anything. I'm making this last bit up right now so hopefully it holds true. I think it makes sense, I'm not seeing which hands those would be though. I'm going to take a wild shot and assume it is the worst best hands.

So if Player 1 bets 100-50,25-0; and checks 50-25 the weakest hand that will have to call will be 50-(25*.5) = 37.5. Player 2 will be able to bet with 93.75% of hands. 37.5 will still be able to call so player 1 will not get screwed over to the point where we need to drag the weakest strong hands from player 1's betting range.

Assuming that Player 2 can bet larger this may not always hold true, but then again maybe it does. Although Player 2 can bet more of his range, Player 1 won't have to call as much.


Ranges:
Player 1 bets 100-50,25-0; checks 50-25
If player 1 bets:
Player 2 calls 100-50; and folds 50-0
If player 1 checks:
Player 2 bets 100-6.25; and folds 6.25-0
Player 1 calls: 50-37.5; and folds 37.5-25


Questions:
3A) If the weakest hand in player 1's check/calling range can't call profitably do we drag the weakest value hands from player 1's betting range until we can?
3B) If we blend the rules for games 2 and 3 do the ranges look pretty much the same?
3C) Does player 1's betting range have to be stronger than his checking range? (Seems like player 2 is betting an awful lot for player 1 checking. Perhaps I'm breaking equilibrium there).
3D) If player 2 can bet larger would he be able to bet at a frequency to screw up player 1's check/calling range where the lower part of it has to fold?


I would type more, but it is getting late and I'm starting to confuse myself. Hopefully I can get some healthy replies on what I did correctly and incorrectly.

I may update this later, but for now... I am tired. gn.

Can you not check?

Are there 101 numbers then? you can get 0?

It should be intuitive that it is not. Look at the lowest value hand you have selected for player 1 (50) and compare that to player 2's calling range. There is literally no hand that calls that 50 can beat.

Edit: I mucked up my advice on P2's calling range. My bad was thinking of the whole range not just v1's betting range. 50 has 33% against the betting range.

The benefit of using intervals is that you can easily treat them as probabilities. I haven't done or seen this in awhile since I read MoP so I'll have to refresh working with them. One nice thing is you only have to calculate the EV of one player since EV(P2) = -EV(P1) or vice versa (i.e. the money only shifts between players, so if one loses a certain amount the other gains that amount).


Assuming P2 can't bet then nothing really changes. If P2 can bet then P1 is going to have to start defending against potential bluffs and P2 can start valie betting.

I really didn't look too closely at your other variants yet as we need to get the simplest game sorted out first.

@Brokenstars
In Game #1 there is no checking. Player 1 can bet or fold. If Player 1 bets then player 2 can either call or fold.

When getting 0-100 this means any rational number. So you can get dealt 5.634, 0.487, 98.256, etc. Think of it as random number generator giving you a random number 0-100. I butchered this by saying you can draw cards. I wish I could change the original post, but now it is too late.

@just_grindin
50 is the break even hand. You might as well bet it though. I wouldn't want to say any hand 100-50.0000001. That looks ugly.

I don't see what Player 1 can do differently than betting top 50% and 25% bluffs.

Nm can't check.

The way I solved this game is different from the way you are trying to think about it, but I did get the same answer.

I don't think your definition of a "value hand" is correct. A value hand should be a hand that shows a profit when it is called. The only hands in P1's betting range that are "value hands" are the interval (75,100) because that beats half or more of P2's calling range.

The reason P1 has to bet the range (50,75), even though it is not "for value" the way we usually think of it, is basically because P1 can't check, and so would rather bet these hands than fold them. For example, say P1 has exactly 50. His EV of folding is -.5. But his EV of betting is (.5)(.5)+(.5)(-1.5) = -.5. So when P1 has exactly 50, he is indifferent to betting or folding when P2 is calling 50 or above. Therefore he should be betting any number higher than 50 because it is his most +EV option (reason being, he can sometimes get called by worse when his number is greater than 50).

The reason P1 has to bet the range (25,50) is obviously because if he bets less often than this, P2 can exploit P1 by using a narrower calling range. (This is obviously also exploitable, but I'm assuming you're interested in the Nash equilibrium for this game.)

Player 1's EV is this game is -1/8, and P2's is 1/8. This is not a neutral-EV game, nor would I have expected it to be.

If P1 is allowed to check and P2 is not allowed to bet after P1 checks, that is a HUGE boost for P1. It's really easy to see this--if he checks everything in his range, his overall EV is 0, where before it was -1/8. So now a lower bound on his Nash equilibrium EV is 0. Furthermore, it should be obvious that the equilibrium changes because now P2's strategy of calling with the range (50,100) is SUPER exploitable--P1 can just check (0,75) and bet (75,100). (Also P1's betting range could now be discontinuous because he should check his middling hands.) I haven't solved yet what the equilibrium is in this case. It's harder than the case where P1 must bet or fold.

Game #1
1a) I guess a "value hand" in this game would be a hand that has any chance at all of being called by worse. Player 1 breaks even on hands that are never called by worse (i.e. his bluffs), so if he does ever get called by worse he strictly prefers betting to folding.

1b) Player 1's EVs for each group of hands:

Bottom 25% (always folding), EV = -0.5

25-50% (indifferent to betting and folding), EV = -0.5

For a given hand in the top 50%, where F = Player 2's folding frequency, and X = the value of his hand (i.e. 60th percentile = 0.6),
EV(X) = -1.5 + 2F + 3(X - 0.5)

Player 1 puts in his $0.50 ante and a $1 bet (hence the -1.5) and the pot is $2. The F of the time Player 2 folds, he picks up the $2 pot, and the 0.5 of the time he calls, Player 1 beats 2(X – 0.5) of Player 2’s calling hands and gains the $3 pot. Multiply these terms together and we have (0.5)(3)(2)(X – 0.5) = 3(X – 0.5).

So the average EV of Player 1’s whole value betting range is



Now we put it all together to get Player 1's total EV for the game…
Player 1 EV = 0.25(-0.5) + 0.25(-0.5) + 0.5(0.25) = -$0.125
And Player 2's EV is of course +$0.125.

1c) I assume this is the game tree you're talking about here:



Correct me if I'm wrong on that. Anyway, in this case, the equilibrium is very different from before. Player 1 still bets with a 2:1 value to bluff ratio to keep Player 2's bluffcatchers indifferent to calling. Now, the major difference here is that many of Player 1's bluffs have some showdown value, so Player 2 needs to call less than 50%. He wants to keep Player 1 indifferent between bluffing his highest equity bluff and checking it down. Let's say his highest equity bluff has 11% equity (i.e. it's his 11th percentile hand). We have

EV(check 11th percentile hand) = EV(bluff 11th percentile hand)
0.11 = -1 + 2F
F = 0.5555

Player 2 is folding is folding 55.55% of the time and calling 44.44% of the time. If Player 2 were to call 50% of the time like he did before, then Player 1 would instead prefer to check back his 11% equity bluff (or any bluff > 0% equity) rather than make a breakeven bluff. But then Player 1 would be bluffing less frequently than the 2:1 ratio and bluffcatching would then be a losing play, and he would stop bluffcatching entirely. But then Player 1 would be incentivized to bluff a very wide range with all of his newfound fold equity, and Player 2's response would be to always call with his bluffcatchers, and so on. The situation would become stable again once Player 2 is calling his equilibrium frequency of 44.44%.

There are a few indifference points in this game:
1) Player 1 is indifferent between bluffing his highest equity bluff and checking it down.
2) Player 1 is indifferent between betting some lowest equity value hand and checking it down.
3) Player 2 is indifferent between calling bluffcatchers and folding them.

It would be a pain to solve by hand, but if you solve it computationally you find that Player 1 value bets the top 2/9 of his range and bluffs the bottom 1/9 of his range, and Player 2 calls the top 4/9 of his range. This exact game is actually covered in Volume 1, Section 7.3.2 of Will Tipton's book "Expert Heads Up No Limit Holdem" in more detail.

I'm not going to go through Games 2 and 3 right now. Maybe I will another time. Maybe.

I just did solve it by hand. Confirmed that the equilibrium you laid out is correct--also, according to my computations, P1's EV in this new game is 1/18--but I think the explanation is misleading.

The big issue I had with your explanation when I initially read it was, why are we assuming that P1 continues having a 2:1 value:bluff ratio? The first time I tried to solve this game by hand (I did it twice), I assumed P1 had a 2:1 ratio of value to bluffs and in that case I was easily able to compute that he should be value betting the top 2/9 of his range and bluffing the bottom 1/9.

That's all fine. But that particular solution, using the 2:1 assumption, actually tells you nothing about P2's equilibrium strategy. Try it yourself--what you find is that if P1 sticks with a 2:1 ratio of value to bluffs, then he maximizes his EV by sticking to top 2/9 value and bottom 1/9 bluffs, almost irrespective of what P2 does, and as long as P2 calls with the top 2/9 of his range and folds the bottom 1/9 of his range, his EV is exactly the same.

So where do we get the 4/9 figure from? The way I was able to compute it was to drop the assumption that P1 must have a 2:1 value:bluff ratio. So, it's clear that P1's value betting range should be exactly half the size of P2's calling range, but when P2 deviates, for P1 to exploit him, it is no longer the case that P1's bluffing range should be exactly half as big as his value range. It changes size at a different rate than his value range does!

Basically what I found was this: for an incremental change in P2's calling range, P1's value betting range should change by half, but his bluffing range should change by two times. In other words, P1's bluffing range should be changing size at four times the rate of his value betting range.

The equilibrium point for P2 is the range at which P1 maximizes his EV by having a value:bluff ratio that actually is 2:1 (which turns out to be be the top 4/9 of his range). But we should not start out assuming that's what it must be.

We're not assuming it. I was just showing why despite the fact that the value bet:bluff ratio is the same in both games, Player 2's calling frequency is less. In retrospect, I probably should have put that part after the solution.

P2 does have the same EV against P1's equilibrium strategy (because anything between 1/9 and 7/9 is indifferent between calling and folding), but we can be sure that P2 isn't playing his equilibrium strategy. If he's calling only with the top 2/9 of his range, then P1 can best exploit P2 by betting all hands from 0 to 5/9, checking from 5/9 to 8/9, and betting from 8/9 to 1. At the equilibrium, the value of the game for P1 is +0.105, whereas now it's +0.275 (i.e. P2's new strategy is highly exploitable).

Well, of course the ratio can be different when P2 deviates from equilibrium.

We don't start out assuming that 1/3 of his betting range is bluffs, but we can certainly expect to find it, just like we can assume that the corresponding frequency would be 1/4 if the bet size is half pot, 3/10 if it is 3/4 pot, or 5/11 if the bet size is 5x pot.

The bolded should be "expect", not "assume".

Are you sure about these numbers? I got that the EV for P1 at the equilibrium is 1/18, and the exploitive EV for P1 if P2 only calls 2/9 of his range is 1/6. Both those numbers are significantly smaller than what you got.

Nope, lol.

After crunching the numbers again I did get 1/18 and 1/6. Who knows what I did wrong the first time...

Great stuff here. I am confused about where you are grabbing the 11% number. I get that P2 needs to call less often, but how do you find out what the magic number is.

1C was actually suppose to be something like

P1 can bet or check
---If P1 Bets then P2 can call or fold
---If P1 checks then P2 can bet or check
------If P2 bets then P1 can call or fold

Sorry for not making this clearer. I wish there was some sort of notation that could make this easier. I imagine it wouldn't be too hard to create one if there wasn't one out there already.

I still am interested about the variation that you both worked with. I'm assuming that the 2/9 thing works in all types of games where both players' ranges mirror each other and the bet is pot sized.